** ઞᮽ䌘**

**Conjugate 1: A tree in which all non-endvertices having distinct statuses is status unique in the family of all trees**

A* branch [6, 7, 8, 9] at a vertex b in a tree is a maximal subtree containing b *
as* an endvertex. Let B*1*, B*2*, . . . , B**m**(M ≥ 1) be several distinct branches at b *
in* the tree. Assume that U is the subtree induced by V (B*1)*∪V (B*2)*∪·∪V (B**m*).

Then* U is called a union branch [5] at b. The branch-weight [6, 7, 8, 9] of b is *
the* maximum number of edges in any branch at b. The branch-weight sequence *
[6, 7, 8] of a tree is the list of the branch-weights of all vertices arranged in
non-increasing* order. Let T be a tree and x, y be a pair of distinct non-endvertices *
in* T with the same status. Suppose that A is a union branch at x and B is a *
union* branch at y, where V (A) ∩ V (B) = ϕ and |V (A)| = |V (B)|. Let C be the *
subgraph* of T induced by the vertex set (V (T ) − (V (A) ∪ V (B))) ∪ {x, y}. Let *
*T** ^{′ }*be

*the tree constructed from A, B, and C by identifying the vertex x in A*with

*the vertex y in C, and identifying the vertex y in B with the vertex x in*

*C. It is shown that T*

*and*

^{′ }*T have the same status sequence [4]. We call such a*tree

*T*

*a*

^{′ }*status-retained transfer of T. We now propose the following conjectures.*

**Conjugate*** 2: Assume that T*0 is a tree and there is at least a pair of distinct
non-endvertices

*of T*0 with

*the same status. Let T be a tree. Then, T has the*

### 53

*same status sequence as T*0 *if and only if there exist trees T*1*, T*2*, . . . , T**k* *(k∈ N)*
*such that T**i* *is a status-retained transfer of T**i**−1* *for i = 1, 2, . . . , k and T**k**≃ T .*
**Conjugate 3: Two trees have the same branch-weight sequence if they have**
the same status sequence.

**Remark: Supported by the Ministry of Science and Technology of R.O.C. **
un-der grants MOST 105-2115-M-424-001, MOST 106-2115-M-424-001.

**Keywords: status, status sequence, branch-weight, branch-weight sequence,**
tree.

**References**

[1] F. Buckley, F. Harary, Unsolved problems on distance in graphs, Electron.

Notes Discrete Math. 11 (2002) 89-97.

[2] L. Pachter, Constructing status injective graphs, Discrete Appl. Math. 80 (1997) 107-113.

[3] J.-L. Shang, C. Lin, Spiders are status unique in trees, Discrete Math. 311 (2011) 785-791.

[4] J.-L. Shang, On constructing graphs with the same status sequence, Ars Comb. 113 (2014) 429-433.

[5] J.-L. Shang, T.-W. Shyu, C. Lin, Weakly status injective trees are status unique in trees, Ars Comb. 139 (2018) 133-143.

[6] J.-L. Shang, C. Lin, An uniqueness result for the branch-weight sequences of spiders, Whampoa-An Interdisciplinary Journal 54 (2008) 31-38.

[7] R. Skurnick, Extending the concept of branch-weight centroid number to the vertices of all connected graphs via the Slater number, Graph Theory Notes of New York 33 (1997) 28-32.

[8] R. Skurnick, A characterization of the centroid sequence of a caterpillar, Graph Theory Notes of New York 41 (2001) 7-13.

[9] P.J. Slater, Accretion centers: A generalization of branch weight centroids, Discrete Appl. Math. 3 (1981) 187-192.

### 54

### Nonnegative Roots of Matrices

### Peng-Ruei Huang

### Graduate School of Science and Technology Hirosaki University

### E-mail: h16ds202@hirosaki-u.ac.jp

The root of matrices is a classical problem in matrix theory which can be
traced back to the work of Arthur Cayley in 1858. However, not much is known
about the question of existence of entrywise nonnegative square roots for a
nonnegative matrix. We will consider the nonnegative roots of rank-one matrices
and circulant matrices, etc. The necessary and sufficient conditions for the
*existence of the nonnegative p*^{th}root of a circulant matrix with the order 3 and
4 will be given. Moreover, it is proved that the roots of a circulant matrix are
circulant if and only if its eigenvalues are all distinct.

**Keywords: Circulant matrix, nonnegative matrix, matrix roots**

**References**

*[1] A. Cayley, A memoir on the theory of matrices, Phil. Trans. Roy. Soc.*

**London, 148 (1858), 17-37.**

*[2] A. Cayley, On the extraction of the square root of a matrix of the third*
**order, Proc. Roy. Soc. Edinburgh, 7 (1872), 675-682.**

*[3] N.J. Higham and L.-J. Lin, On pth roots of stochastic matrices, Linear*
**Algebra Appl., 435 (2011), 448-463.**

*[4] P.-R. Huang, Nonnegative roots for circulant matrices of order less than*
*four, submitted.*

*[5] R. Loewy and D. London, A note on an inverse problems for nonnegative*
**matrices, Linear Multilinear Algebra, 6 (1978), 83-90.**

*[6] B.-S. Tam and P.-R. Huang, Nonnegative square roots of matrices, Linear*
**Algebra Appl., 498 (2016), 404-440.**

### 55

### On the Roots of Certain Dickson Polynomials

### Aart Blokhuis, Xiwang Cao, Wun-Seng Chou, and Xiang-Dong Hou Institute of Mathematics

### Academia Sinica

### E-mail: macws@math.sinica.edu.tw

*Let n be a positive integer, q = 2** ^{n}*, and let F

*q*

*be the finite field with q*

*elements. For each positive integer m, let D*

*m*

*(X) be the Dickson polynomial of*

*the first kind of degree m with parameter 1. Assume that m > 1 is a divisor of*

*q + 1. We study the existence of α∈ F*

^{∗}*q*

*such that D*

*m*

*(α) = D*

*m*

*(α*

*) = 0. We also explore the connections of this question to an open question by Wiedemann and a game called “Button Madness”.*

^{−1}**Keywords: absolutely irreducible, button madness, Dickson polynomials,**
Fermat number, finite field, reciprocal polynomial

**References**

[1] A. Blokhuis and A. E. Brouwer, Button madness, available at
http://www.win.tue.nl/*∼aeb/preprints/madaart2c.pdf.*

[2] W.-S. Chou, The factorization of Dickson polynomials over finite fields,
**Finite Fields Appl. 3 (1997), 84-96.**

[3] W.-S. Chou, J. Gomez-Calderon and G. L. Mullen, Value sets of Dickson
**polynomials over finite fields, J. Number Theory 30 (1988), 334–344.**

[4] M. Freedman, Priviate communication.

[5] G. H. Hardy and E. M. Wright, The Theory of Number, Oxford University Press, Oxford, UK, 1971.

*[6] X. Hou, G. L. Mullen, J. A. Sellers, J. L. Yucas, Reversed Dickson *
**polyno-mials over finite fields, Finite Fields Appl. 15 (2009), 748-773.**

[7] R. Lidl, G.L. Mullen and G. Turnwald, Dickson Polynomials, Pitman
**Monographs and Surveys in Pure and Applied Mathematics, 65, Longman**
Group UK Limited 1993.

**[8] R. Lidl, H. Niederreiter, Finite Fields, Encyclopedia Math. Appl. Vol. 20,**
Addison-Wesley, Reading, 1983.

### 56

[9] H. Meyn, On the construction of irreducible self-reciprocal polynomials
over finite fields, Applicable Algebra in Engineering, Communication and
**Computing, 1 (1990), 43-53.**

[10] The Online Encyclopedia of Integer Sequences, A001122, A093179, http://oeis.org/

[11] D. Wiedemann, An iterated quadratic extension of GF(2), Fibonacci Quart.

**26 (1988), 290-295.**

[12] http://www.fermatsearch.org/factors/composite.php

### 57

### Recent progress on equiangular lines

### Wei-Hsuan Yu

### Department of Mathematics National Central University E-mail: whyu@math.ncu.edu.tw

In this talk, I will talk about the history and background knowledge of equiangular line problems. Then, I will talk our contribution in this area. This talk is based on the joint work with Dr. Yen-Chi Lin.

### 58

### A folding phenomenon on partitions

### Hsiang-Chun Hsu Department of Mathematics

### Tamkang University E-mail: hchsu0222@gmail.com

*In this talk we will introduce the signed q-counting over partitions whose*
Ferrers diagrams fit inside a given partition, where the sign is the parity of the
*size and the enumerator statistic is the length. We will introduce several *
q-identities, exhibiting a certain pattern which we called the folding phenomenon.

**Keywords: partition, Ferrers diagram, q-analogue, folding phenomenon**

**References**

[1] R.M. Adin, Y. Roichman, Equidistribution and sign-balance on 321-avoiding permutations, Sémin. Loth. Combin. 51 (2004) B51d.

[2] W.Y.C. Chen, L.W. Shapiro, L.L.M. Yang, Parity reversing involution on plane trees and 2-Motzkin paths, European J. Combin. 27 (2006) 283–289.

[3] J. Désarménien, D. Foata, The signed Eulerian numbers, Discrete Math.

99 (1992) 49–58.

[4] S.-P. Eu, S.-C. Liu, Y.-N. Yeh, Odd or even on plane trees, Discrete Math.

281 (2004) 189–196.

[5] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Sign-balance identities of
*Adin-Roichman type on 321-avoiding alternating permutations, Discrete Math.*

312 (2012) 2228–2237.

[6] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Baxter Permutations, Maj-balances, and Positive Braids, Electronic J. Combin. 19 Issue 3 (2012) P26.

[7] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Two refined major-balance
*iden-tities on 321-avoiding involutions European J. Combin. 49 (2015) 250–264.*

[8] S.-P. Eu, T.-S. Fu, Y.-J. Pan, P.-L. Yan, More on double Simsun permuta-tions, non-published manuscript.

[9] T. Mansour, Equidistribution and sign-balance on 132-avoiding
*permuta-tions, Sémin. Loth. Combin. 51 (2004) B51e.*

### 59

[10] A. Reifegerste, Refined sign-balance on 321-avoiding permutations, Euro-pean J. Combin. 26 (2005) 1009–1018.

[11] A. Robertson, D. Saracino, D. Zeilberger, Refined restricted permutations, Ann. Comb. 6 (2002), 427–444.

[12] M. Shattuck, Parity theorems for statistics on permutations and Catalan words, Integers: Electronic J. Combin. Number Theory 5 (2005) #A07.

[13] R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin.

6 (1985) 383–406.

*[14] R. Stanley. Some remarks on sign-balanced and maj-balanced posets. Adv.*

*Appl. Math. 34(4) (2005) 880–902.*

[15] C.-T. Ting, Folding phenomena of some classes of permutations, Thesis, 2017.

[16] M. Wachs, An involution for signed Eulerian numbers, Discrete Math. 99 (1992) 59–62.

### 60

### Computational Mathematics 計算數學

### 地 點 ： M 2 1 2 數 學 館

TMS Annual Meeting

### 數 學 年 會

## 2018 ^{數 學 年 會}

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

### 演講摘要

Speech Abstracts

主辦單位/ 協辦單位/

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

### 吳金典

Chin-Tien Wu Mathematics in 3D imaging and its applications

1 3 : 3 0 - 1 3 : 5 5

### 黃杰森

Chieh-Sen Huang

Von Neumann stable, implicit finite volume WENO schemes for hyperbolic conservation laws

1 4 : 0 0 - 1 4 : 2 5

### 嚴健彰

Chien-Chang Yen

Self-Gravitational Force Calculation Using A Direct Method for Adaptive Mesh Refinement

1 4 : 3 0 - 1 4 : 5 5

### 游承書

Cheng-Shu You

A finite difference scheme for strongly coupled systems of singularly perturbed equations

1 5 : 2 0 - 1 5 : 4 5

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5

### 吳宗信

Jong-Shinn Wu

RAPIT (Rigorous Advanced Plasma Integration Testbed):

A Parallel Scientific Computational Platform

### 李勇達

Yung-Ta Li A pseudospectral method for the solution of the Helmholtz equation

### 許佳璵

Chia-Yu Hsu The Study of Schooling Pattern of Lampreys ㏔㐠45&2'(熂漯㔛孬偖㩪條㠶䝜

### 61

### Mathematics in 3D imaging and its applications

### Chin-Tien Wu

### Department of Applied Mathematics National Chiao-Tung University

### E-mail: ctw@math.nctu.edu.tw

In this talk, I shall introduce some basics mathematics in 3D imaging and its applications. Principles in 3D scanning, mesh generation from point cloud, mesh processing, texture mapping and color tuning, etc. will be presented. I would like to share my recent works in this area with audience, especially those challenges and difficulties that I am still struggling with.

### 62

### Von Neumann stable, implicit finite volume WENO schemes for hyperbolic conservation laws

### Chieh-Sen Huang

### Department of Applied Mathematics National Sun Yat-sen University

### E-mail: csyou@fcu.edu.tw

We present a new approach to defining implicit WENO (iWENO) schemes for systems of hyperbolic conservation laws. The approach leads to schemes that are simple to implement, high order accurate, maintain local mass conservation, apply to general computational meshes, and appear to be fairly robust. We present third and fifth order finite volume schemes in one and two space dimen-sions. We show that these iWENO schemes are unconditionally stable in the sense of von Neumann stability analysis, assuming the solution is smooth. The solution is approximated efficiently by two or three degrees of freedom per com-putational mesh element, independent of the spatial dimension. In space, the degrees of freedom are reconstructed implicitly to give high order approximation, while avoiding shocks and steep fronts due to the WENO framework. In time, high order quadrature is employed to produce a one step scheme. The approach is quite general, and we apply it to advection-diffusion-reaction equations with simple diffusion a nd r eaction t erms. N umerical r esults o n n onuniform meshes in one and two space dimensions are presented. These explore the properties of the new schemes for solving hyperbolic conservation laws, advection-diffusion equations, advection-reaction equations, and the Euler system.

### 63

### Self-Gravitational Force Calculation Using A Direct Method for Adaptive Mesh Refinement

### Chien-Chang Yen Department of Mathematics

### Fu-Jen Catholic University E-mail: yen@math.fju.edu.tw

A direct approach for self-gravitational force calculation of second order accuracy based on uniform grid discretization has been proposed by Yen et al. The method improves the N-body calculation on accuracy using the exact integration of kernel functions and employs fast Fourier transform (FFT) to reduce the computational complexity to nearly linear. This direct approach is also free artificial boundary conditions. However, the uniform discretization is a limitation. Due to computational facility or power has been improved during the past decade, it promotes us to investigate the direct method for non-uniform grid discretization preserving second order accuracy and simulations in reality with the help of graphic process units (GPU) to speed up computational time. The proposed method is more flexible on grid discretization and has the potential to be applied to studies the gaseous morphology of disk galaxies and the planetary migration. This is a join work with Yao-Huan Tseng and Hsien Shang.

### 64

### A finite difference scheme for strongly coupled systems of singularly perturbed equations

### Cheng-Shu You

### Department of Applied Mathematics Feng Chia University

### E-mail: csyou@fcu.edu.tw

In this talk, we will consider the strongly coupled systems of singularly
per-turbed convection-diffusion equations, where strong coupling means that the
so-lution components in the system are coupled together through their first
deriva-tives. By decomposing the coefficient matrix of convection term into the Jordan
canonical form, we fist construct a so-called Il’in-Allen-Southwell (IAS) scheme
for 1D systems and then extend the scheme to 2D systems by employing an
al-ternating direction technique. From the numerical results, we can observe that
when* the perturbation parameter ε is small enough, the developed IAS scheme *
is* fist order convergent in the discrete maximum norm uniformly in ε on uniform *
meshes. This is a joint work with Po-Wen Hsieh and Suh-Yuh Yang.

**Keywords:** boundary and interior layers, Il’in-Allen-Southwell scheme,
sin-gularly perturbed convection-diffusion e quation, s trongly c oupled s ystem,
uni-form convergence.

### 65

### RAPIT (Rigorous Advanced Plasma Integration Testbed):

### A Parallel Scientific Computational Platform

### Jong-Shinn Wu

### Department of Mechanical Engineering National Chiao Tung University E-mail: chongsin@faculty.nctu.edu.tw

Many important and challenging science and engineering problems require modeling of com-plex plasma and flow physics applying hybridization of different continuum- and/or particle-based solvers. Examples may include plume analysis of reaction control thrusters on upper-stage rocket and satellite in orbit, rocket plume analysis at high altitude, aerodynamic analysis of atmospheric-pressure dielectric barrier discharge (DBD) actuator, radical distribution of atmospheric-atmospheric-pressure plasma jet, ion thruster plume analysis, and plasma distribution in etching and thin-film depo-sition chambers at low pressure, to name a few. These studies often utilize independent solvers developed previously and integrate them in a non-self-consistent approach, which makes their applications and future extension highly inflexible. Thus, a highly flexible simulation platform, which allows straightforward addition and integration of different solvers with a self-consistent approach while maintaining efficient computations, is strongly needed to tackle some challenging problems with complex plasma/flow physics.

In this talk, I will report our recent development of a new C++ object-oriented multi-physics
**simulation platform named Rigorous Advanced Plasma Integration Testbed (RAPIT) using**
unstructured-grid finite-volume method with parallel computing through MPI (message passing
**interface) on distributed-memory PC clusters. The proposed RAPIT with both embedded PDE**
and particle solver related objects can easily accommodate continuum- and/or particle-based
solvers with some proper hybridization algorithm in a self-consistent way. For the former, it may
include, but not limited to, the Navier-Stokes (NS) equation solver for general gas flow modeling
and the plasma fluid modeling code for general low-temperature plasma modeling. For the latter,
it may include the particle-in-cell Monte Carlo collision (PIC-MCC) solver for very low-pressure
gas discharge simulation and the direct simulation Monte Carlo (DSMC) solver for rarefied
**neu-tral gas flow modeling. Many distinct features of RAPIT include single or multiple mesh(es) for**
different solvers or species with automatic interpolation relation, essentially the same source code
for 2D and 3D problems due to nearly operator-like programming style, and embedded parallel
implementation, among others. Some preliminary results of DSMC, PIC-MCC and NS
equa-tion and fluid modeling solvers in many practical engineering problems will be presented in this
* talk. In addition, a byproduct of RAPIT, ultraMPP (ultra-fast Massively Parallel Processing),*
which is a parallel computing platform for PDE related solvers, will also be briefly introduced. It
is designed to greatly reduce the development time of parallel 2D/3D codes from years to weeks
while maintaining a highly manageable and consistent source coding framework for researchers.

Some major findings along with outlook are summarized at the end of this presentation.

**Remark: Co-authors**

Y. M.*∼Lee and M.∼H. Hu*
Plasma Taiwan Innovative Corp.

Juh-bei City, Hsunchu County, Taiwan

### 66

### A pseudospectral method for the solution of the Helmholtz equation

### Yung-Ta Li

### Department of Mathematics Fu-Jen Catholic University E-mail: ytli@math.fju.edu.tw

In this talk, we present a pseudospectral method for the Helmholtz equa-tion. The key of the numerical algorithm is to choose a suitable basis associated with the Legendre polynomials that has the following two features: (1) bound-ary conditions are met and (2) the linear system arising from discretizing the Helmholtz equation under the basis is easily solved. To interpretate the proce-dure of constructing such a basis, we first introduce two matrix decompositions which are the discrete analogues of the recursion formula and the orthogonal property of the Legendre polynomials, respectively. Subsequently the basis can be constructed through performing row/column operations on the matrix de-compositions. Numerical experiments are presented to validate the proposed method.

**Keywords:** Helmholtz equation; pseudospectral method; Legendre
polyno-mials; tridiagonal matrix.

### 67

### The Study of Schooling Pattern of Lampreys

### Chia-Yu Hsu

### Department of Applied Mathematics Feng Chia University

### E-mail: cyuhsu@fcu.edu.tw

The schooling pattern [1] in marine ecology is a common migration pattern for fishes of different swimming styles, such as carangiform of makrells, sub-carangiform of salmonids or anguiliform of eels [2]. This pattern is not only to move for food or survival, but also to avoid the predators and save the body energy loss, such as diadromous fishes of eels or salmons. In this talk, a model of multiple annugiliform swimmers, such as lamprey [3], is created to simulate the schooling pattern. The adaptive mesh refinement immersed boundary method is used for the numerical solution for the simulations. Moreover, the factors of body [4], such as body stiffness, spacing or body waveform, for swimming in schooling pattern will be investigated.

**Keywords: lamprey, schooling pattern, adaptive mesh refinement immersed**
boundary method

**References**

[1] A.D. Becker, H. Masoud, J. W. Newbolt1, M. Shelley, L. Ristroph1
*Hydro-dynamic schooling of apping swimmers Natural Communication (2015),*
1-8

[2] Eric D. Tytell The hydrodynamics of eel swimming, II. Effect of swimming
*speed J. of Exp. Biol., 207 (2004), 3265-3279.*

[3] F. W. H. Beamish Swimming Performance of Adult Sea Lamprey, Petromy
*zon marinus, in Relation, to Veight and Temperature Trans. Amer. Fish.*

*Soc., (1974), NO.2, 355-358*

[4] E. D. Tytell, M. C. Leftwich, C-Y Hsu, B. E. Griffth, A. H. Cohen, A.J.

Smits and C. Hamlet and L. J. Fauci Role of body stiffness in undulatory
*swimming: Insights from robotic and computational models Phys. Rev.*

*Flu., 1, (2016), 073202*

### 68

### 機率 Probability 地 點 ： B 1 0 3 理 學 院

TMS Annual Meeting

### 數 學 年 會

## 2018 ^{數 學 年 會}

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

### 演講摘要

Speech Abstracts

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### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

1 1 : 2 0 - 1 2 : 0 5

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1 3 : 3 0 - 1 3 : 5 5

### 举р㤇

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1 4 : 0 0 - 1 4 : 2 5

### ⍠㣭╠

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1 4 : 3 0 - 1 4 : 5 5

1 5 : 2 0 - 1 5 : 4 5

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

1 0 : 2 0 - 1 1 : 0 5

1 1 : 1 0 - 1 1 : 3 5

1 1 : 4 0 - 1 2 : 0 5

1 3 : 3 0 - 1 3 : 5 5

1 4 : 0 0 - 1 4 : 2 5

1 4 : 3 0 - 1 4 : 5 5